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PHYSICAL REVIEW E 88, 023010 (2013)
Transition to chaos of natural convection between two infinite differentially heated vertical plates
Zhenlan Gao*
CNRS, LIMSI, UPR3251, BP 133, 91403 Orsay Cedex, France andUniversite Pierre et Marie Curie-Paris 6, 4, Place Jussieu, 75252 Paris, Cedex 05, France
Anne SergentCNRS, LIMSI, UPR3251, BP 133, 91403 Orsay Cedex, France and
Universite Pierre et Marie Curie-Paris 6, 4, Place Jussieu, 75252 Paris, Cedex 05, France
Berengere Podvin†
CNRS, LIMSI, UPR3251, BP 133, 91403 Orsay Cedex, France
Shihe XinCETHIL, INSA de Lyon, 69621 Villeurbanne Cedex, France
Patrick Le QuereCNRS, LIMSI, UPR3251, BP 133, 91403 Orsay Cedex, France
Laurette S. TuckermanPMMH, ESPCI, 75005 Paris, France
(Received 15 April 2013; published 12 August 2013)
Natural convection of air between two infinite vertical differentially heated plates is studied analytically intwo dimensions (2D) and numerically in two and three dimensions (3D) for Rayleigh numbers Ra up to 3 timesthe critical value Rac = 5708. The first instability is a supercritical circle pitchfork bifurcation leading to steady2D corotating rolls. A Ginzburg-Landau equation is derived analytically for the flow around this first bifurcationand compared with results from direct numerical simulation (DNS). In two dimensions, DNS shows that therolls become unstable via a Hopf bifurcation. As Ra is further increased, the flow becomes quasiperiodic, andthen temporally chaotic for a limited range of Rayleigh numbers, beyond which the flow returns to a steadystate through a spatial modulation instability. In three dimensions, the rolls instead undergo another pitchforkbifurcation to 3D structures, which consist of transverse rolls connected by counter-rotating vorticity braids.The flow then becomes time dependent through a Hopf bifurcation, as exchanges of energy occur between therolls and the braids. Chaotic behavior subsequently occurs through two competing mechanisms: a sequence ofperiod-doubling bifurcations leading to intermittency or a spatial pattern modulation reminiscent of the Eckhausinstability.
DOI: 10.1103/PhysRevE.88.023010 PACS number(s): 47.55.P−, 47.20.Ky, 47.20.Bp, 47.27.Cn
I. INTRODUCTION
Transition to turbulence of natural convection in a fluidlayer between two differentially heated plates is of substantialinterest for many industrial applications, such as heat exchang-ers in reactors or insulation of buildings (e.g., double-panedwindows). In classic Rayleigh-Benard convection, the fluidlies between two horizontal plates and is heated from belowso the thermal gradient opposes the direction of gravity. Inthe configuration considered here, the walls are vertical, andthe thermal gradient is orthogonal to the direction of gravity.Although this configuration may not have received quite asmuch attention as Rayleigh-Benard convection [1], a numberof studies have shed some light on its specific dynamics.However, with a few exceptions, most of the theory andnumerics so far have focused either on the two-dimensionalcase or on a limited range around the critical Rayleigh number.
*[email protected]†[email protected]
The first investigations of natural convection betweendifferentially heated vertical plates relied on experimentaldescriptions. Elder [2] observed the onset of secondary flowand “cat’s-eyes” tertiary flow in his experiments with twofluids: paraffin and silicone oil. Vest and Arpaci [3] observedthe onset of secondary convection in air flow and comparedit with their calculations of secondary flow stream patterns.Oshima [4] considered convection in a rectangular water-filledcavity heated through two vertical sidewalls. He observed thedevelopment of the wavy motions of streamlines into rows ofperiodic vortices, which then burst into turbulence.
Early computations of the flow structure were based onstability analysis in the neighborhood of the critical Rayleighnumber Rac and were limited to the 2D case. FollowingBatchelor’s [5] approach, Gill and Davey [6] and then Bergholz[7] investigated the 2D linear stability of the flow for differentPrandtl numbers. Vest and Arpaci [3] also relied on 2D linearstability to calculate secondary-flow stream patterns. Weaklynonlinear stability calculations in 2D were carried out byDaniels and Weinstein [8], Cornet and Lamarque [9] forair, and Mizushima and Gotoh [10] for water convection. A
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GAO, SERGENT, PODVIN, XIN, LE QUERE, AND TUCKERMAN PHYSICAL REVIEW E 88, 023010 (2013)
Newton-Raphson method to compute equilibria was used byMizushima and Saito [11] in the zero-Prandtl-number limit.More bifurcation diagrams were obtained in the case of air byMizushima and Tanaka [12,13].
The agreement between the patterns predicted by theory andexperimental observations of the first instabilities suggests thatconvection between 2D infinite plates is a good representationof what happens in tall cavities. However, there is someevidence that these simplified assumptions may not provide anaccurate picture of the real dynamics. First, the influence of theaspect ratio—what exactly makes a cavity “tall enough”—maybe difficult to determine intuitively. From Bergholz [7], it canbe seen that the nature of the most unstable disturbances forwater (Pr = 7.5) switches from a stationary to a traveling statefor a critical aspect ratio A ∼ 70. Moreover, the nature ofthe higher-order bifurcations is likely to be affected by thepresence of horizontal boundaries. S. Xin [14] carried out 2Dnumerical simulations of natural convection of air in a confinedcavity as well as between infinite plates (a channel). Transitionto the chaotic state was observed at a relatively low Rayleighnumber in the cavity, while only regular patterns could beobtained in the channel.
Second, although most numerical simulations [15–20] havebeen carried out in the 2D case, there is evidence thatthe flow becomes rapidly three-dimensional as the Rayleigh(or Grashof) number increases, as was observed by Wrightet al. [21] in their experiments for tall air-filled cavities.This is in agreement with Chait and Korpela’s [22] 3Dlinear stability calculations, which indicated that the flowbetween vertical isothermal sidewalls should become rapidlythree-dimensional. Early 3D computations of equilibria inthe limit of vanishing Prandtl number were performed byNagata and Busse [23]. A steady 3D pattern was found to beassociated with a secondary instability. Clever and Busse [24]also identified a steady 3D pattern, which then bifurcated into atraveling wave of invariant shape at higher Rayleigh numbers.A Ginzburg-Landau model was used by Suslov and Paolucci[25,26] to describe the three-dimensional flow between infiniteplates in the case of non-Boussinesq convection for a variety ofPrandtl numbers. However, their approach remains limited to aregion around the critical Rayleigh number. Bratsun et al. [27]carried out extensive experimental and numerical studies ofthe successive bifurcations of the flow in three dimensions, buttheir work was conducted at a high Prandtl number (Pr = 26)for which the primary instability consists of traveling waves.No equivalent study has been performed for air as far as weknow.
At the other end of the Rayleigh number range, the turbulentregime has received some attention for two decades. Phillips[28] compared 3D direct numerical simulation results withthe experiments of Elder [29] and showed that most of theturbulence was generated by the shear layer at the center ofthe slot. Versteegh and Niewstadt [30,31] computed energybudgets to determine scaling laws and wall functions witha direct application to turbulence modeling. Although thepresence of spiral structures has been noted by Wang et al. [32]for air at Ra = 5.4 × 105, a complete coherent-structure-baseddescription is still missing. As noted by Hall [33], “there hasbeen apparently no attempt to look for coherent structuresembedded in the flows as has become routine in shear flows.”
Few models for the dynamics of such structures have yet beenproposed [33].
The goal of the present study is to provide a descriptionof the transition to turbulence of natural convection of airbetween two infinite plates. We are aware of the limitedcharacter of this investigation, which crucially depends onthe periodic dimensions of the plates. Due to numericalconstraints, the transverse dimension of the plates was keptsmall, which hampers the development of three-dimensionalinstabilities. However, we believe that such simulations con-stitute a necessary step towards a better understanding of thedynamics of unsteady natural convection. They also representa complementary approach to linear stability analysis, wherethe full range of wave numbers can be explored. A detailedanalysis of the flow at lower Rayleigh numbers could behelpful to understand the coherent structures observed in thefully developed turbulent regime [32]. Some direct numericalsimulation (DNS) studies [30] suggest that a substantial partof the energy in the turbulent regime is associated withpatterns which are similar to the most linearly unstable mode.This situation presents some analogy with Rayleigh-Benardconvection, where the large-scale structures identified in theturbulent regime share common features with the convectioncells observed at near-critical Rayleigh numbers [1,34,35].In addition, determining key instability mechanisms in acanonical configuration will be useful for studying morecomplex geometries and/or including additional physics suchas radiation or mixed convection.
The paper is organized as follows. We first recall standardtheoretical results based on linear and weakly nonlinearstability analysis. We derive analytically a Ginzburg-Landauequation to represent the flow around the first bifurcation. Wethen briefly present results for 2D direct numerical simulationfor Rayleigh numbers Ra up to about 3Rac. We show that atemporally chaotic regime can only be found over a limitedrange of Rayleigh numbers. We then present our results ofa 3D numerical simulation. The sequence of the instabilitiesleading to chaos and corresponding physics is described indetail. A large difference with 2D results is found.
II. PHYSICAL AND NUMERICAL CONFIGURATION
We consider the flow of air between two infinite verticalplates maintained at different temperatures. The configurationis represented in Fig. 1(a). The distance between the two platesis D, and the periodic height and depth of the plates are Lz
and Ly , respectively. The temperature difference between theplates is set to �T . The x direction is normal to the plates, y
represents the transverse direction, and the gravity g is oppositeto the vertical direction z.
The relevant fluid properties are the kinetic viscosity ν,thermal diffusivity κ , and thermal expansion coefficient β.Four nondimensional parameters characterize the flow: thePrandtl number Pr = ν/κ , the respectively transverse andvertical aspect ratios Ay = Ly/D and Az = Lz/D, and theRayleigh number based on the gap between the two platesRa = gβ�T D3
νκ. Only the Rayleigh number dependence is
considered in the present study. The Prandtl number is fixedand equal to 0.71. The transverse aspect ratio is set to beAy = 1. The vertical aspect ratio was Az = 10 in most cases
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0 0.5 1
−0.4
−0.2
0
0.2
0.4
0.6
X
W(x
)
0 0.5 1−0.5
0
0.5
X
Θ(x
)
(a) (b)
FIG. 1. (Color online) (a) Study domain. (b) Base flow profiles for the temperature �(x) and vertical velocity W (x).
and occasionally Az = 9 and Az = 2.5. The choice of thesevalues is dictated by the critical wave number kc ∼ 2π/2.5 (seenext section). The larger domain therefore could accomodatefour structures.
A. Equations of motion
The flow is governed by the Navier-Stokes equationswithin the Boussinesq approximation. We choose the referenceparameters to be κ
D
√Ra for the velocity, D for the length, and
�T for the temperature. The nondimensionalized equationsthen are
�∇ · �u = 0, (1)
∂ �u∂t
+ �u · �∇ �u = −�∇p + Pr√Ra
��u + Pr θ z, (2)
∂θ
∂t+ �u · �∇ θ = 1√
Ra�θ, (3)
with Dirichlet boundary conditions at the plates,
�u(0,y,z,t) = �u(1,y,z,t) = 0,(4)
θ (0,y,z,t) = 0.5, θ (1,y,z,t) = −0.5,
and periodic conditions in the y and z directions. Here t denotestime, �u = (u,v,w) is the velocity vector, p is the pressure, andθ is the temperature.
The equations of motion (1)–(4) admit an analytic steadysolution (U ,V ,W ,�), the pure conduction state, which dependsonly on the x direction,
U = 0; V = 0;
W (x) = 16
√Ra
[(x − 1
2
)3 − 14
(x − 1
2
)]; (5)
�(x) = −(x − 1
2
).
The solution Eq. (5) is represented in Fig. 1(b). The equa-tions (1)–(4) admit an O(2) × O(2) symmetry. One O(2) sym-
metry corresponds to the translations in the transverse directiony and the reflection y → −y, while the other corresponds tothe translations in the vertical direction z and a reflectionthat combines centrosymmetry and Boussinesq symmetry:(x,z,T ) → (1 − x,−z,−T ). The base flow possesses thesame symmetry as the equations, since it is one-dimensionaland antisymmetric with respect to the midplane x = 0.5.
B. Numerical configuration
The 2D simulations are carried out using a spectral DNScode [14] developed at the Laboratoire d’Informatique pour laMecanique et les Sciences de l’Ingenieur (LIMSI). The coderelies on a Chebyshev collocation discretization in the hori-zontal direction x and a Fourier discretization in the verticaldirection z. The complete Navier-Stokes system is solved byinverting the Uzawa operator to ensure incompressibility. Overthe range of Rayleigh numbers investigated, we choose 40Chebyshev modes for the discretization in x and 160 Fouriermodes in z for Az = 10.
The 3D simulations are carried out with a multidomainspectral code [36] also developed at LIMSI. A Chebyshev-Fourier collocation method is used for spatial discretization.Incompressibility is enforced by the projection-correctionmethod. The equations are integrated in time with a second-order mixed explicit-implicit scheme. The domain decomposi-tion is carried out by the Schur complement and implementedwith the MPI library. Some details about the numericalschemes are explained in Appendix A. In our study, the domainis decomposed into four subdomains in the z direction. AChebyshev discretization is applied in directions x and z, whilea Fourier discretization is used in the transverse direction y.For Ay = 1 and Az = 10, we use 30 modes in the transversedirection y and 40 and 160 modes in the horizontal andvertical, x and z, directions, respectively. The initial conditionis typically taken to be equal to the base flow.
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Convergence of the spatial discretization in the 2D sim-ulation has been established [14]. Convergence for the 3Dcode was checked by running the simulations using 45 Fouriermodes in y and 60 and 240 Chebyshev modes in x and z,respectively. No significant difference was observed betweenthe two spatial discretizations. Comparison of the 2D and 3Dcodes was carried out both below and slightly above the criticalRayleigh number. The difference between the flows was foundto be within round-off error.
III. 2D THEORETICAL ANALYSIS OF THEFIRST BIFURCATION
A. Linear stability analysis
The base flow Eq. (5) is parallel and depends only on thex direction. The hypotheses of Squire’s theorem are verifiedin this case, so the most unstable mode is expected to be 2D.We decompose the velocity and temperature into the base flow(U,W,�) and perturbations (u,w,θ ) as
u = U + u, w = W + w, θ = � + θ, (6)
and take the curl of the momentum equations to eliminatethe pressure term. Let ψ be the stream function for theperturbations, where u = − ∂ψ
∂z, w = ∂ψ
∂x. The equations of
motion (1)–(4) lead to the 2D system of perturbations in x
and z as
M∂φ
∂t= Lφ + b(φ,φ), (7)
with
φ =[
ψ
θ
]; b =
[bψ
bθ
]; M =
[∇2 00 1
];
(8)
L =⎡⎣ Pr√
Ra∇4 − W ∂
∂z∇2 + ∂2W
∂x2∂∂z
Pr ∂∂x
∂�∂x
∂∂z
1√Ra
∇2 − W ∂∂z
⎤⎦ ,
where the nonlinear terms bψ and bθ are bilinear forms anddefined as
bψ (φα,φβ) =(
∂ψα
∂z
∂
∂x− ∂ψα
∂x
∂
∂z
)∇2ψβ
(9)
bθ (φα,φβ) =(
∂ψα
∂z
∂
∂x− ∂ψα
∂x
∂
∂z
)θβ.
The indices α and β will be used to designate the differentorders of the solutions obtained in the multiscale analysis inthe following subsection.
The associated boundary conditions are
ψ(x = 0) = ψ(x = 1) = 0, (10)
ψ ′(x = 0) = ψ ′(x = 1) = 0, (11)
θ (x = 0) = θ (x = 1) = 0, (12)
where the prime symbol denotes partial differentiation withrespect to x.
The linearized system is written as
M∂φ
∂t= Lφ. (13)
Seeking a normal-mode solution of the form φ = φ(x)est+ikz,we obtain a generalized eigenvalue problem, Lφ = sMφ.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
X
|φ(x
)|
|ψ(x)||θ(x)|
FIG. 2. (Color online) Norm of the most unstable mode |φ(x)|(|ψ(x)|,|θ (x)|) at kc and Rac.
Solving the eigenvalue problem leads to a critical Rayleighnumber Rac = 5708 and a critical wave number kc = 2.81,which agrees with the results of Bergholz [7] and Ruth [37].The critical eigenvalue is purely real, and the modulus of themost unstable mode |φ(x)| at the critical wave number kc andRayleigh number Rac is maximum in the core region, as shownin Fig. 2.
B. Weakly nonlinear analysis
We use a multiscale analysis to derive a Ginzburg-Landauequation for the flow around the first bifurcation. We introducethe asymptotic expansion of the perturbation to the baseflow as
φ = εφ1 + ε2φ2 + ε3φ3 + O(ε4). (14)
We also define several time scales, t0 = t,t1 = εt,t2 = ε2t , andvertical length scales z, z0 = z,z1 = εz. We then expand allthe operators in system (7) with respect to the new variables,
M = M0 + εM1 + ε2M2 + O(ε3), (15)
L = L0 + εL1 + ε2L2 + O(ε3), (16)
b = b0 + εb1 + ε2b2 + O(ε3), (17)
where M0, M1, M2, L0, L1, L2, b0, and b1 are detailed inAppendix B.
The first-order perturbation εφ1 can be expressed as
εφ1 =[
ψ
θ
]= A(t1,t2,z1)
[ψ(x)θ(x)
]est+ikcz + c.c., (18)
where ψ,θ are the most unstable modes at the wave numberkc given by the linear stability analysis, A is the amplitude ofthe solution, and c.c. stands for complex conjugate.
Substituting these expansions into (7), and collecting theterms at different orders of ε, yields problems at differentorders of ε. The problem at order ε coincides with the linearstability analysis. Collecting terms at order ε2 and imposinga compatibility condition gives an expression for the groupvelocity Cg . Numerical evaluation of Cg yields a value ofabout 10−5, which is close to the expected value of zero sincethe rolls are steady. The discrepancy is likely to be due to thediscretization error.
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1. Ginzburg-Landau equation
Collecting terms at order ε3 and enforcing the solvabilitycondition leads to a Ginzburg-Landau equation [38] for theamplitude A in the primitive variables,
∂A
∂t= σ (Ra − Rac)A + γ
∂2A
∂z2− lA2A∗. (19)
Numerical evaluation of the coefficients σ , γ , l forthe critical wave number kc = 2.81 gives σ = 7.67 × 10−5,γ = 0.112, l = 20.45. The sign of l indicates that the bifurca-tion is supercritical. The amplitude of perturbations predictedby this Ginzburg-Landau equation will be compared with theDNS results discussed in the next section.
2. Absolute instability
The linearized Ginzburg-Landau equation takes the form
∂A
∂t= σ (Ra − Rac)A + γ
∂2A
∂z2. (20)
The introduction of a particular solution in the form ofnormal modes A = Aei(βz−ωt) into Eq. (20) results in thedispersion relation:
D(ω,β,Ra) = σ (Ra − Rac) + iω − γβ2 = 0. (21)
It can be written in the form of a single temporal modeω(β,σ,Ra) = i[σ (Ra − Rac) − γβ2], which has an equilib-rium (β0,ω0), satisfying the conditions ω0 = ω(β0) and∂ω∂β
(β0) = 0. The latter condition yields ∂ω∂β
(β0) = −2iγβ0 = 0.So β0 = 0 and ω0 = iσ (Ra − Rac). As ω0,i = σ (Ra − Rac) > 0,the disturbance grows with time at any fixed station in thelaboratory frame, which corresponds to the absolute instability,in agreement with Tao and Zhuang’s [39] results.
IV. NUMERICAL RESULTS
A. 2D DNS simulations
We first use a 2D simulation to study the development ofinstabilities in the flow. Results are summarized in Table I.
1. First bifurcation
As predicted by the linear stability analysis, the baseflow bifurcates to four steady corotating rolls at Rac ∼ 5708.Although vertical invariance is broken, the solution stilldisplays the symmetry D4, consisting of translation by theheight Az/4 of each of the rolls. Invariance of the equa-tions under z translations ensures that there exists a wholecircle of solutions, corresponding to an arbitrary verticaltranslation of the rolls: The bifurcation is a circle pitchforkbifurcation.
The time evolution of the temperature measured at the point(x = 0.0381, z = 5), located in the hot boundary layer, isplotted in Fig. 3(a). An enlargement of the same signal forthe times 1500 < t < 2000 is represented in logarithmic scalein Fig. 3(b). The temperature disturbance grows exponentiallyfor 1500 < t < 1750, which corresponds to the linear growthof the most unstable eigenmode and then increases at a slowerrate for t > 1750 before the amplitude of the solution saturates.As was pointed out by Henderson and Barkley [40], thisevolution shows that the coefficient of the cubic term in thenormal form of the circle pitchfork bifurcation is negative,and, therefore, the bifurcation is supercritical, in agreementwith the prediction of the Ginzburg-Landau model.
The steady amplitude A computed from the Ginzburg-Landau equation is compared with the amplitude of thevelocity and temperature perturbations observed in the DNSfor a domain which is as close as possible to the criticalwavelength. The periodic height of the DNS was adjustedto Az = 9, so it featured a wavelength of λ = 2.25 close to thecritical wavelength λc = 2.236.
Figures 4(a) and 4(b) show that the agreement between theGinzburg-Landau model and the DNS is very good for boththe temperature and velocity up to Ra ∼ 6300 (about 10%above Rac).
2. Subsequent bifurcations
Figure 5 shows how the spatial organization of the flowvaries with increasing Ra. Just above the critical Rayleighnumber, the flow is characterized by four steady structures,as shown in Fig. 5(a) at Ra = 6000. As Ra is increased past
TABLE I. Summary of bifurcations and associated flow structures and symmetries for the 2D simulations, Az = 10.
Nature of Spatial features Spatial symmetryRa bifurcation Number of structures Temporal symmetry
Ra < Rac = 5708 1D base flow O(2)Steady
Rac � Ra � 13 000 Supercritical Corotating rolls D4
Circle pitchfork n = 4 Steady13 500 � Ra � 15 300 Supercritical Corotating rolls No symmetry
Hopf n = 3 Periodic15 400 � Ra � 15 600 Corotating rolls No symmetry
Unknown n = 3 Quasiperiodic15 700 � Ra � 17 000 Corotating rolls No symmetry
Unknown n = 3 “Chaotic”18 000 � Ra � 21 000 Corotating rolls No symmetry
Unknown n = 2 Steady
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0 1000 2000 30000
0.5
1
1.5
2x 10−3
time
ΔT
1400 1600 1800 200010
−6
10−5
10−4
10−3
10−2
time
ΔT
(a) (b)
FIG. 3. (Color online) (a) Time series of temperature perturbation at point (0.0381, 5) in the boundary layer near the hot wall at Ra = 6000.(b) An enlargement of (a) for 1500 < t < 2000 on a logarithmic scale.
the value of Ra = 13 500, the four steady rolls merge intothree rolls which oscillate in time, as shown in Fig. 5(b). Atstill higher Rayleigh numbers Ra � 18 000, only two rolls ofunequal size are observed, as shown in Fig. 5(d).
The temporal spectrum of the vertical velocity at thepoint (x = 0.0381, z = 0.683) is shown in Fig. 6(a) and ischaracterized by a main frequency (with harmonics) of f1 =0.032. When 15 000 � Ra � 16 000, the flow still consists ofthree oscillatory rolls, but the temporal evolution of the flowbecomes more complex. When Ra is increased to Ra = 15 500,the flow becomes quasiperiodic with the appearance of asecond, much lower, frequency, f2 = 0.0031 [Fig. 6(b)]. WhenRa = 16 000, the peaks around the main frequency f1 and f2
broaden [see Fig. 6(c)], corresponding to seemingly chaoticbehavior. The chaotic behavior subsides beyond Ra = 18 000,as the flow spatial pattern is modified: There seems to bea competition between the onset of purely temporal chaosin a specific flow pattern and the development of spatialinstabilities at short wave numbers. Up to Ra = 21 000, whichwas the highest Rayleigh number considered, the flow remainssteady with a robust two-roll pattern.
B. 3D DNS results: First bifurcation
We first check that the base flow remains stable with respectto any perturbation when Ra < Rac. As expected, the firstbifurcation observed in the DNS occurs at Rac around 5800and is characterized by the appearance of four 2D steadycorotating rolls which are represented in Fig. 7. As mentionedabove, the vertical translation invariance is replaced with a D4
symmetry, and the centro-Boussinesq symmetry is conserved.
C. Second bifurcation: 3D steady structures
When Ra > Rac2, the four-roll solution becomes unstablein the transverse direction, and a steady 3D pattern, shownin Fig. 8, appears through a second bifurcation, as was alsofound by Nagata and Busse [23] and Clever and Busse [24].Since the transition breaks the y-translation invariance, thisnew bifurcation is also a circle pitchfork bifurcation. From theanalysis of a time series similar to that of Fig. 3, we concludethat this bifurcation is supercritical.
The threshold Rac2 can be obtained by linearizing theequations of motion around the 2D steady solution and
5800 6000 62000
1
2
3
4
5x 10−3
Ra
θ
Ginzburg−LandauDNS
5800 6000 62000
0.01
0.02
0.03
0.04
0.05
Ra
w
Ginzburg−LandauDNS
(a) (b)
FIG. 4. (Color online) Comparison of the maximum amplitudes observed in the 2D DNS at x = 0.0381 with the Ginzburg-Landau equationprediction: (a) temperature and (b) vertical velocity.
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X
Z
0 0.5 1
2
4
6
8
10
X
Z
0 0.5 1
2
4
6
8
10
X
Z
0 0.5 1
2
4
6
8
10
XZ
0 0.5 1
2
4
6
8
10
(a) Ra=6000 (b) Ra=13 500 (c) Ra=15 000 (d) Ra=18 000
FIG. 5. 2D flow streamlines at different Ra. (a) Four steady coro-tating rolls; [(b) and (c)] three oscillating rolls; (d) two steady rolls.
integrating a small perturbation in time for different values ofRa. The growth rate of these perturbations, which is equal tothe most unstable eigenvalue, is found to increase quasilinearlywith Ra. The critical Rayleigh number Rac2 obtained by linearextrapolation of the plot is around 9980, which is within 2%of the value Ra ∼ 10 100 observed in the DNS.
The steady 3D solution retains some of the symmetry ofthe 2D solutions, namely the reflection in y and translation byAz/4 in z, but the translation symmetry in y and the centro-Boussinesq symmetry are replaced with the single discretesymmetry
(x,y,z,T ) → (1 − x,y + 0.5,Az − z,−T ). (22)
The 2D solution, which was O(2) × D4 symmetric, hasbifurcated to a 3D solution with D1 × D4 symmetry.
This can be seen in Fig. 9, which shows temperaturecontours and streamlines on three planes parallel to theplates. The field obeys the symmetry (22), as can be seenby comparing Figs. 9(a) with 9(c) or Figs. 9(d) with 9(f). Theupwind motion on the plane x = 0.0381 along the hot wall[Fig. 9(d)] and the downwind motion on the plane x = 0.9619along the cold wall [Fig. 9(f)] are captured. On the midplane,the streamline plot of Fig. 9(e) shows two large and two small
Y
X
Z
(a) (b) (c)
FIG. 7. (Color online) Flow structure at Ra = 6000. (a) Isocon-tours of the temperature on the two selected vertical plates x = 0.0245and y = 0.9677, (b) isosurface of transverse vorticity �y = 3.1, and(c) enlargement of the upper half of the domain in (b).
secondary counter-rotating vortices. Since the flow is invariantunder translation by Az/4, we restrict our analysis to the upperhalf of the domain in the rest of this section.
Figure 10 shows isosurfaces of the vorticity components�y , �x , �z along with the Q criterion [41]. The Q criterion isdefined as
Q = 12 (�i�i − EijEij ),
where Eij is the rate of strain tensor 12 ( ∂ui
∂xj+ ∂uj
∂xi) and therefore
provides a measure of the vortices. Comparison of Figs. 10(a)and 10(d) show that most of the vorticity is transverse and isorganized into the corotating convection rolls correspondingto the most linearly unstable mode.
Examination of the horizontal vorticity (�x) plots inFig. 10(b) confirms that the flow is characterized by twocounter-rotating vortices, which are inclined 45◦ with respectto both the horizontal and the vertical planes. As shown inFig. 10(d), these secondary circulations link the primary rollsand are to some extent reminiscent of the three-dimensionalbraids connecting the primary vortices observed in shear layers
0 0.05 0.10
0.01
0.02
0.03
f0 0.05 0.1
0
0.01
0.02
0.03
0.04
0.05
f0 0.05 0.1
0
0.002
0.004
0.006
0.008
0.01
f
(a) Ra = 15 000 (b) Ra = 15 500 (c) Ra = 16 000
FIG. 6. (Color online) Temporal Fourier spectrum of the vertical velocity at the point (0.0381, 0.683) in the boundary layer near the hotwall in 2D simulations for different Rayleigh numbers.
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Y
X
Z
(a) t=500 (b) t=1040 (c) t=1090 (d) t=1120 (e) t=1170 (f) t=2000
FIG. 8. Flow structure at Ra = 11000. [(a)–(f)] Temperature isocontours on the two planes x = 0.0245 (next to the hot wall) and y = 0.9677(perpendicular to sidewalls) at times as indicated.
before vortex pairings [42]. In addition, Fig. 10(c) shows thepresence of additional counter-rotating vortices within theprimary convection rolls. These vortices are about halfthe height of the larger secondary vortices, and their orientationis opposite to that of the larger secondary vortices. They arepredominantly aligned with the vertical direction z.
In summary, the flow structure for the 3D pattern consistsof (1) principal vertical corotating rolls predominantly alignedin the transversal direction y, (2) large secondary counter-rotating vortices or braids linking up the primary rolls, whichare inclined about 45◦ with respect to the horizontal and thevertical planes, and (3) two vertical, short counter-rotatingvortices located within the principal rolls.
D. Third bifurcation: 3D time-periodic flow
The 3D pattern remains stable up to a value of Ra < Rac3.For Rac3 < Ra < 12 000, the 3D pattern becomes time depen-dent.
1. Local analysis
The time series of the temperature at a point located in thehot boundary layer is plotted in Fig. 11(a). Figure 11(b) showsthat it corresponds to a periodic signal of frequency f1 =0.036, very close to the basic frequency f 2D
1 = 0.032 foundin the 2D simulations at a slightly higher Rayleigh number.
During this periodic regime, we observe that the oscillationfrequency f1 is nearly constant as the Rayleigh number
Y
Z
0 0.50
2
4
6
8
10
Y
Z
0 0.50
2
4
6
8
10
Y
Z
0 0.50
2
4
6
8
10
Y
Z
0 0.50
2
4
6
8
10
Y
Z
0 0.50
2
4
6
8
10
Y
Z
0 0.50
2
4
6
8
10
(a) x = 0.0381 (b) x = 0.5 (c) x = 0.9619 (d) x = 0.0381 (e) x = 0.5 (f) x = 0.9619
FIG. 9. [(a)–(c)] Temperature isocontours on two planes which are symmetric with respect to the vertical midplane (x = 0.5); [(d)–(f)]streamlines on the same three vertical planes at Ra = 11 000.
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Y
X
Z
(a) Ωy = 3 (b) Ωx = ±1 (c) Ωz = ±0.5 (d) Q = 0.12
FIG. 10. (Color online) Vorticity isosurfaces at Ra = 11 000.
increases above its bifurcation value, while the square ofthe oscillation amplitude increases linearly with Ra. Thisis consistent with a Hopf bifurcation. The critical Rayleighnumber Rac3 evaluated by the linear extrapolation of theoscillation amplitude as a function of Ra is around Rac3 =11 270, in excellent agreement with results from the DNSobservations.
2. Global enstrophy budgets
Some insight into the dynamics of the flow can be given byenstrophy, which gives a measure of rotational effects in theflow. The total contribution to the enstrophy of each vorticitycomponent [
∫V
�2j dV ]1/2 was computed, where j = x,y,z
and �j is the j -th component of the vorticity. As we cansee from Fig. 12, most of the enstrophy is contained in thetransverse contribution [
∫V
�2ydV ]
12 . Both the horizontal and
vertical contributions oscillate essentially in phase oppositionwith the transverse contribution.
A physical interpretation of this plot is given in Fig. 13.The flow structures are similar to the steady ones observedin the previous regime (and therefore retain the same spatialsymmetry), but they now pulse periodically. When the primary
2000 2020 2040 20600
2
4
6
8
10
12
time
[V
Ω2xdV ]
12
[V
Ω2ydV ]
12
[V
Ω2zdV ]
12
[V
(Ω2x + Ω2
y + Ω2z)dV ]
12
FIG. 12. (Color online) Temporal evolution of horizontal vortic-ity intensity [
∫V
�2xdV ]
12 , transverse vorticity intensity [
∫V
�2ydV ]
12 ,
and vertical vorticity intensity [∫
V�2
zdV ]12 at Ra = 11 500.
rolls are strongest, the secondary vortices disappear. At thismoment, the flow is mostly two-dimensional [Fig. 13(c)]. Incontrast, when the secondary vortices reach their maximumintensities, the primary rolls bend in the transverse direction;the strongly 3D flow can be seen in Fig. 13(e).
We note that the frequency f of the oscillation is very closeto the natural frequency of the mixing layer fn ∼ 0.032 [42],when it is nondimensionalized with the distance between theplates and the maximum velocity difference observed in thebase flow.
To better understand the origin of the oscillations, weconsider the vorticity equation:
∂�i
∂t+ uj
∂�i
∂xj
− �j
∂ui
∂xj
= Pr√Ra
∂2�i
∂xj∂xj
− Pr εijk
∂θδj3
∂xk
.
(23)
If we multiply equation (23) by 2�i , we obtain
∂�2i
∂t+ uj
∂�2i
∂xj
= 2�i�j
∂ui
∂xj
+ 2Pr√Ra
�i
∂2�i
∂xj∂xj
− 2�iPr εijk
∂θδj3
∂xk
. (24)
(We sum over j and k but not over i, and ε is the permutationsymbol and δ is the Kronecker symbol.) The terms on the
0 500 1000 1500 20000
0.001
0.002
0.003
0.004
0.005
time
ΔT
0.05 0.1 0.15 0.2 0.2510
−2
10−1
100
101
f
T
(a) (b)
FIG. 11. (Color online) (a) Time series of temperature perturbation at the point (0.0381, 0.097, 5) in the boundary layer near the hot wall,Ra = 11 500. (b) Temporal Fourier spectrum of the periodic portion t ∈ [1200,2000] of the signal (a).
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right-hand side of the equations correspond to theproduction—or destruction—of �2
i through three differentmechanisms: (i) vortex stretching, which is tilting and stretch-ing of vorticity components by the velocity field, (ii) friction,i.e., the action of viscosity (which we will also refer to asdiffusion), and (iii) buoyancy. Summing over i (i.e., using
the tensor notation for i) yields the enstrophy equation. Wechoose instead to integrate the equation corresponding toeach component over the domain. One can check that thetransport term (second term on the left-hand side) disappears,so we are left with the following equations for each vorticitycomponent:
⎡⎢⎢⎣
∫V
∂�2x
∂tdV∫
V
∂�2y
∂tdV∫
V
∂�2z
∂tdV
⎤⎥⎥⎦ =
⎡⎢⎢⎣
∫V
(2�x( �� · �∇u) + 2 Pr√
Ra�x∇2�x + 2Pr�x
∂θ∂y
)dV∫
V
(2�y( �� · �∇v) + 2 Pr√
Ra�y∇2�y − 2Pr�y
∂θ∂x
)dV∫
V
(2�z( �� · �∇w) + 2 Pr√
Ra�z∇2�z
)dV
⎤⎥⎥⎦ . (25)
Figures 14 to 16 present the global evolution and spatialdistribution of the three terms on the right-hand side of Eq. (39)for each vorticity component. The balance in the horizontaldirection can be seen in Fig. 14(a). The vortex-stretchingterm and the buoyancy term are both source terms andoscillate with a small phase shift, while the friction term isnegative and oscillates in phase opposition with the othertwo contributions. As the intensity of the friction is not quitecompensated by the effect of vortex stretching and buoyancy,this results in limited oscillations of the horizontal vorticityrms [
∫V
�2xdV ]
12 . The spatial distribution of the contributions
due to vortex stretching, viscous effects, and buoyancy isrepresented in Figs. 14(b)–14(d) on a plane orthogonal to theplates. The plane was chosen in order to provide a relevantcross section of one of the vorticity braids, i.e., the secondarycounter-rotating vortices. The vortex-stretching and buoyancyterms are both positive everywhere [Figs. 14(b) and 14(d)],while the friction term is negative [Fig. 14(c)]. All terms reachtheir maximum over the portion of space occupied by thecounter-rotating vortices.
Figure 15(a) represents the different contributions toequation (25) for the transverse component. All three termsoscillate essentially in phase (with phase shifts of about 1/8and 1/12 of the time period), which is responsible for thestrong oscillation observed in the principal rolls. The vortex-
(a) t = 2001 (b) t = 2008 (c) t = 2015 (d) t = 2022 (e) t = 2029
FIG. 13. (Color online) Q criterion isosurface Q = 0.12 atselected times (corresponding vertical lines in Fig. 12) spanning onetemporal oscillation, Ra = 11 500.
stretching term is always positive, as can be expected. Perhapsa more surprising result is that friction is now a source term forthe transverse vorticity, while buoyancy constitutes a sink forit. Since the temperature gradient is always negative and theprincipal rolls are associated with positive transverse vorticity�y , one would expect a positive value for −2Pr �y
∂θ∂x
. To un-derstand this discrepancy, we examined the spatial distributionof the different contributions, which can be seen in Figs. 15(b)–15(d) for the symmetry plane y = 0.5. The effect of vortexstretching was essentially positive, as could be expected[Fig. 15(b)]. Over the portion of space covered by the principalvortices, the contribution of the buoyancy was also found tobe positive [Fig. 15(d)], but strongly negative values wereobserved very close to the wall in the boundary layer, whichis where the temperature gradient is significant. The situationwas reversed for friction effects: strongly positive values werefound very close to the walls. This reflects the fact that trans-verse vorticity is indeed generated at the walls through friction,while buoyancy works against the velocity gradient in the walllayer.
Figure 16(a) represents the relative contributions of vortexstretching and friction to the oscillations of the verticalcomponent of the enstrophy. These oscillations are limited,since the positive effect of vortex stretching is almost exactlycompensated by frictional effects (buoyancy does not appearin the equations). As can be seen in Figs. 16(b) and 16(c), bothfriction and vortex-stretching contributions are maximal at thelocation of the counter-rotating vortices.
E. 3D subsequent bifurcations
1. Period-doubling bifurcations
Case Az = 10. When 12 100 � Ra � 12 200, the tempo-ral evolution of the 3D pattern becomes more complex.At Ra = 12 200, the time series of the temperature at apoint located in the boundary layer presents subharmonicoscillations for t ∈ [800,2200] before becoming quite irreg-ular, as shown in Fig. 17(a). The Fourier spectrum of thetemperature [Fig. 17(b)] shows that the largest amplitudeis located at the frequency f1 = 0.035, which is close tothe frequency identified in the previous periodic regime ata slightly lower Rayleigh number (see Sec. IV D1), whilethe second-largest amplitude corresponds to the frequencyf1/2 = 0.0175 = f1/2.
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2000 2020 2040 2060
−5
0
5
10
time
∂Ω2x
∂tvortex stretchingviscous effectsbuoyancy
X
Z
0 0.5 10
2
4
6
8
10
3210
-1-2-3
VortexStretching
X
Z
0 0.5 10
2
4
6
8
10
3210
-1-2-3
ViscousEffects
X
Z
0 0.5 10
2
4
6
8
10
3210
-1-2-3
Buoyancy
)d()c()b()a(
FIG. 14. (Color online) (a) Temporal evolution of the different terms in Eq. (25) for the x component: time derivative of the transverse
vorticity∫
V
∂�2x
∂tdV , vortex-stretching contribution
∫V
2�x( �� · �∇w)dV , and friction effects∫
VPr√Ra
�x∇2�xdV ; [(b)–(d)] spatial distribution
of (b) the vortex-stretching contribution 2�x( �� · �∇u), (c) the friction contribution Pr√Ra
�x∇2�x , and (d) the buoyancy contribution 2Pr �x∂θ
∂y
on the vertical plane y = 0.2903 at t = 2000, Ra = 11 500.
The topology of the flow consists of four 3D structureswhich are similar to those found in the monoperiodic regimeat Ra = 11 500 (Sec. IV D1). The intensities of the transverserolls and that of the braids oscillate out of phase, with atemporal modulation equal to twice the basic period.
Case Az = 2.5. Period doubling persists when the heightwas set to Az = 2.5 instead of Az = 10. As Ra increases,
we observe for Az = 2.5 a succession of period-doublingbifurcations as illustrated by phase portraits in Fig. 18. At Ra =12 000, the singly periodic regime is characterized by one cyclein the phase portrait [Fig. 18(a)]. For Ra ∈ [12 100,12 200], aperiod-2 cycle is observed in Fig. 18(b). At Ra = 12 300, weobserve a 4-cycle in the phase portraits and then at Ra = 12 310an 8-cycle and at Ra = 12 320 a 16-cycle [Figs. 18(c)–18(e)].
2000 2020 2040 2060−10
−5
0
5
10
time
∂Ω2y
∂tvortex stretchingviscous effectsbuoyancy
X
Z
0 0.5 10
2
4
6
8
10
151050
-5-10-15
VortexStretching
X
Z
0 0.5 10
2
4
6
8
10
151050
-5-10-15
ViscousEffects
X
Z
0 0.5 10
2
4
6
8
10
151050
-5-10-15
Buoyancy
)d()c()b()a(
FIG. 15. (Color online) (a) Temporal evolution of the different terms in equation (25) for the y component: time derivative of the transverse
vorticity∫
V
∂�2y
∂tdV , vortex-stretching contribution
∫V
2�y( �� · �∇w)dV , and friction effects∫
VPr√Ra
�y∇2�y dV ; [(b)–(d)] spatial distribution
of (b) the vortex-stretching contribution 2�y( �� · �∇v), (c) the friction effects Pr√Ra
�y∇2�y , and (d) the buoyancy contribution −2Pr �y∂θ
∂xon
the vertical plane y = 0.5 at t = 2000, Ra = 11 500.
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2000 2020 2040 2060
−2
0
2
4
6
time
∂Ω2z
∂tvortex stretchingviscous effects
X
Z
0 0.5 10
2
4
6
8
10
3210
-1-2-3
VortexStretching
X
Z
0 0.5 10
2
4
6
8
10
3210
-1-2-3
ViscousEffects
)c()b()a(
FIG. 16. (Color online) (a) Temporal evolution of the different terms in Eq. (25) for the z component: time derivative of the vertical vorticity∫V
∂�2z
∂tdV , vortex-stretching contribution
∫V
2�z( �� · �∇w)dV , and friction effects∫
VPr√Ra
�z∇2�z dV ; [(b) and (c)] spatial distributions of
(b) the vortex-stretching contribution 2�z( �� · �∇w) and (c) the friction contribution Pr√Ra
�z∇2�z at the vertical plane y = 0.2903 (same as inFig. 14) at t = 2000, Ra = 11 500.
At Ra = 12 400, there is no longer evidence of periodicity asthe trajectory fills out the space in a seemingly chaotic fashion[Fig. 18(f)]. Similar sequences of period-doubling bifurcationshave been observed in the experimental transition to chaosin Rayleigh-Benard convection. Maurer and Libchaber [43]observed the appearance of a first frequency f ′
1, followed by asecond frequency f ′
2. For higher values of the Rayleigh num-ber, phase locking between the frequencies was observed. Thetransition to turbulence was then triggered by the generation ofthe frequencies f ′
2/2, f ′2/4, and so forth. A similar scenario was
found in the experiments of Giglio, Musazzi, and Perini [44],where a reproducible sequence of period-doubling bifurcationsup to f ′
1/16 was observed.
As the Ra is further increased beyond Ra = 12 550,intermittency is observed in Fig. 19(a). The 3D structures keeppulsating chaotically, but, at random times, the primary rollsdisappear entirely and reform at a different location, which isseparated from the original location by half a wavelength, asshown in Figs. 19(b) and 19(c). This behavior suggests theexistence of a heteroclinic connection between two chaoticattractors, which are diametrically located on the O(2) × O(2)invariant torus of chaotic solutions. The jump from onepulsating structure to another appears to take place onlyin the vertical direction (the system is severely constrainedin the spanwise direction). Structurally stable heteroclinicconnections between fixed points or periodic solutions have
0 1000 2000 3000−0.015
−0.01
−0.005
0
0.005
0.01
time
ΔT
0.05 0.1 0.15 0.2 0.25
10−1
100
101
f
T
(a) (b)
FIG. 17. (Color online) (a) Time series of temperature perturbation at the point (0.0381, 0.097, 5) in the boundary layer near the hot wall,Ra = 12 200, Az = 10. (b) Temporal Fourier spectrum of the subharmonic portion in the time interval t ∈ [1200,2000].
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0.42 0.43 0.44 0.45 0.460.43
0.44
0.45
0.46
0.47
Tb
Ta
0.42 0.43 0.44 0.45 0.460.43
0.44
0.45
0.46
0.47
Tb
Ta
00221=aR)b(00021=aR)a(
0.42 0.43 0.44 0.45 0.460.43
0.44
0.45
0.46
0.47
Tb
Ta
0.42 0.43 0.44 0.45 0.460.43
0.44
0.45
0.46
0.47
Tb
Ta
01321=aR)d(00321=aR)c(
0.42 0.43 0.44 0.45 0.460.43
0.44
0.45
0.46
0.47
Tb
Ta
0.42 0.43 0.44 0.45 0.460.43
0.44
0.45
0.46
0.47
Tb
Ta
00421=aR)f(02321=aR)e(
FIG. 18. (Color online) Phase portraits at different Ra, Az = 2.5. Abscissa: Ta temperature mesured at the point (0.038, 0.097, 0.983);ordinate: Tb temperature measured at the point (0.038, 0.903, 0.983).
been shown to exist in systems with O(2) symmetry [45,46].We are not aware of corresponding theoretical results forchaotic attractors.
2. Development of a spatial instabilityin the large domain Az = 10
For Ra � 12 200, we observed irregular oscillations inFig. 17 for large times t > 2200. This corresponds to adrastic change in the spatial organization of the flow, asshown in Fig. 20. One of the structures is weakened and thendisappears so at large times t > 2200, the pattern observedtypically consists of three structures, as can be seen inFigs. 20(b), 20(c), and 20(f). However, four structures canstill be found intermittently [Figs. 20(a), 20(d), and 20(e)].
The spatial organization of the flow can be described by the1D vertical Fourier transform T (0,k) of the x-averaged tem-perature on the vertical plane y = 0.5. The temporal evolutionof the spectral coefficients |T (0,k)|2 for the modes k = 3,4 isshown in Figs. 21(a) and 21(b). The mode k = 4 dominateswhen 700 < t < 2000, and then for t ∈ [4000,6000] the modek = 3 dominates for most of the time, except around thetimes t ∼ 5040 and t ∼ 5950, where the mode k = 4 becomesdominant again. Due to the long integration times, we werenot able to determine whether the spatial intermittency was atransient or a persistent feature of the flow.
A simulation was performed at Ra = 12 000 from an initialcondition consisting of a instantaneous field at Ra = 12 200characterized by a three-structure pattern. The flow settled
023010-13
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7500 8000 8500 90000.42
0.43
0.44
0.45
0.46
0.47
time
T
X
Z
0 0.5 10
0.5
1
1.5
2
2.5
X
Z
0 0.5 10
0.5
1
1.5
2
2.5
(a) (b) (c)
FIG. 19. (Color online) (a) Time series of the temperature at the point (0.0381, 0.097, 0.983) in the boundary layer near the hot wall atRa = 12 600. [(b) and (c)] Streamlines on the plane Y = 0 at two arrow-pointed instants in (a): (b) t = 8505 and (c) t = 8540.
down to a periodic pulsation of three structures. The presenceof hysteresis confirms that the spatial wave-number modula-tion instability is subcritical and supports the conjecture thatthe wave-number competition between mode 4 and mode 3 issimilar to a subcritical Eckhaus instability.
Beyond Ra = 13 000, for Az = 10, it is no longer possibleto identify a discrete set of frequencies, and the flow rapidlybecomes temporally chaotic. When Ra is increased to Ra =15 000, the whole domain is still dominated by three structures,but patterns of two or four structures can also be observed, asevidenced by the evolution of the temperature spectral density|T (0,k)|2 for different wave numbers.
X
Z
0 0.5 10
2
4
6
8
10
X
Z
0 0.5 10
2
4
6
8
10
X
Z
0 0.5 10
2
4
6
8
10
X
Z
0 0.5 10
2
4
6
8
10
X
Z
0 0.5 10
2
4
6
8
10
X
Z
0 0.5 10
2
4
6
8
10
(a) (b) (c) (d) (e) (f)
FIG. 20. Flow streamlines on the plane y = 0 at Ra = 12 200 fordifferent times: (a) t = 2905; (b) t = 2920; (c) t = 5900; (d) t = 5910;(e) t = 5960; (f) t = 5970.
F. Influence of flow structures on global heat transfer
The global heat transfer is evaluated by the Nusselt number,which is defined as a ratio between the convective anddiffusive heat transfer. In our simulations, the Nusselt numberis calculated as Nu = ∫∫ − ∂ ˜〈θ〉
∂xdydz|x=0,1, since the velocity
of the flow at the walls is zero, where 〈·〉 denotes the timeaveraging of a variable. Its dependence with respect to theRayleigh number Ra is plotted in Fig. 22.
In the 2D steady regime, we find that Nu ∼ 0.0867Ra0.25,which is consistent with the laminar regime. This estimateactually holds slightly beyond the second supercritical pitch-fork bifurcation, where the 2D rolls become more intense anddistorted in the transversal direction (this stage is labeled as“quasi-2D structures” in Fig. 22). However, at Ra = 10 500,the flow becomes three-dimensional through the creation ofsecondary vortices, and the Nusselt number experiences asmall decrease. It then remains approximately constant overthe 3D steady regime from Ra = 10 500 and Ra = 10 600.At the onset of the oscillatory regimes, the Nusselt numberbegins to increase and continues doing so over the sequence ofperiod-doubling bifurcations. After a sharp decrease observedat the onset of the Eckhaus-like instability, the Nusselt numberstarts increasing again. The maximum heat transfer increasesover the range of Rayleigh numbers Ra < 15 000 is about 20%,which agrees with Wright et al.’s [21] results.
We note that the spatial characteristics of Wright et al.’s“secondary cells” match those of what we call primaryrolls. Furthermore, their general description of the route toturbulence seems to agree loosely with ours, as their flowbecomes three-dimensional and then chaotic at a Rayleighnumber of 13 600, which is close to our observations. However,two discrepancies are observed: (i) unlike our stationary rolls,their cells appear to drift in the vertical direction from the onsetof the first instability, which could be due to (horizontal) endeffects of the cavity, and (ii) as the Rayleigh number increases,the motion of the cells is intensified, and merging between thecells occurred, whereas no vortex pairing was observed in oursimulation. This difference could be the consequence of therelatively small dimensions of our numerical domain.
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0 1000 2000 3000−2
0
2
4
6
8
10
12x 10−3
time
| T(0
,k) |2
k=3k=4
4000 4500 5000 5500 60000
1
2
3
4
5x 10−3
time
|T(0
,k)|2
k=3k=4
(a) (b)
FIG. 21. (Color online) Temporal evolution of the spectral coefficients |T (0,k)|2 on the midplane y = 0.5 for selected modes k.[(a) and (b)] Ra = 12 300, modes k = 3,4: (a) t ∈ [0,3000] and (b) t ∈ [4000,6000].
V. CONCLUSION
The focus of the present study is the sequence of instabilitiesleading to chaos for the air flow between two infinitedifferentially heated vertical plates. Our goal is to examine theinfluence of three-dimensional effects in transition. A mappingof the route to chaos has been established in two and inthree dimensions. In both cases the base flow bifurcates to 2Dsteady rolls through a supercritical circle pitchfork bifurcationat Rac = 5708. The nature of this first instability is foundto be absolute. A weakly nonlinear analysis results in thederivation of a Ginzburg-Landau equation which is able topredict correctly the amplitude of the 2D rolls for Rayleighnumbers within a limited range (10%) of Rac.
In 2D simulations, a second bifurcation occurs atRa = 13 500. The flow becomes oscillatory, and the steadyfour-roll pattern turns into a periodic three-roll one with acharacteristic frequency f = 0.032. When Ra is increased,the temporal evolution of the three unsteady rolls becomesquasiperiodic, and then apparently chaotic, while the char-acteristic frequency f remains dominant. As Ra is furtherincreased to Ra = 18 000, the flow becomes steady again, andthe three oscillatory rolls give way to two steady rolls. Thissuggests that the occurrence of pure temporal chaos is limited
0.6 0.8 1 1.2 1.4 1.6x 10
4
1
1.05
1.1
1.15
1.2
1.25
Ra
Nu
steady 2D rollssteady steady 2D structuressteady 3D structuresmono−periodic 3D structuresspatially modulated 3D structures
FIG. 22. (Color online) The Nusselt number Nu (averaged oververtical planes and time) as a function of Ra for Az = 10.
by the development of a vertical instability, which leads to along-wavelength modulation of the spatial pattern. The twosteady rolls remain stable over a range of Rayleigh numbers,as no chaotic behavior is observed up to Ra = 21 000.
The situation differs in 3D simulation (Table II). Thesecond bifurcation is observed at a Rayleigh number ofRac2 ∼ 9980. The 2D rolls become unstable through anothersupercritical pitchfork bifurcation to a steady 3D pattern,characterized by secondary counter-rotating vortices con-necting the principal convection rolls. When the Rayleighnumber is further increased to Rac3 ∼ 11 270, the steady3D pattern becomes oscillatory through a Hopf bifurcation,as the intensities of the transverse rolls and the counter-rotating vortices oscillate in phase opposition. A sequenceof period-doubling bifurcations is then observed at higherRayleigh numbers. If the periodic height of the plates is smallenough (Az = 2.5), thereby preventing any modulation of thebasic vertical wavelength, the period-doubling bifurcationslead to temporal chaos at Ra = 12 400. The location of thechaotically pulsating structures is observed to be intermittentbeyond Ra > 12 550. For larger domains such as Az = 10, thesequence of period-doubling bifurcations is also observed, butonly as a transient feature. The multiply periodic flow givesway to complex spatiotemporal dynamics when Ra � 12 100and a competition between different wavelengths is rapidlyapparent in the flow pattern. The global behavior of heattransfer is established up to Ra = 15 000, where Nu generallyincreases with Ra, with discontinuities as the flow goes throughbifurcations.
Comparison of 2D and 3D results confirms that transverseeffects are essential in the development of instabilities andthe onset of chaos. This result is of interest, and as in manysituations involving thermal convection, the first step towardsmaking a problem tractable is to reduce it to a two-dimensionalgeometry. We emphasize that, due to numerical constraints, thedimensions of the plates were relatively small in the presentstudy. The competition between vertical and transverse patternmodulations is expected to be altered as the dimensions ofthe plate increase. Comparison of the present study withexperimental results highlights the need for larger simulation
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TABLE II. Summary of bifurcations and associated flow structures and symmetries for 3D simulations, Az = 10, T = 1/f1.
Nature of Flow structures Spatial symmetryRa bifurcation Number of structures Temporal symmetry
Ra < Rac = 5708 1D base flow O(2) × O(2)Steady
Rac < Ra < 9980 Supercritical 2D corotating rolls O(2) × D4
Circle pitchfork n = 4 Steady9980 < Ra < 11 270 Supercritical 3D structures D1 × D4
Circle pitchfork n = 4 Steady11 270 < Ra � 12 000 Supercritical 3D structures D1 × D4
Hopf n = 4 T periodicRa � 12 100 Period doubling 3D structures D1 × D4
n = 4 2nT periodicsubcritical Eckhaus-like 3D structures No
instability n = 3 Symmetry
domains, as well as for a better experimental characterizationof transverse effects.
ACKNOWLEDGMENT
Some of the computations were carried out at CNRS-IDRISProject DARI0326.
APPENDIX A: NUMERICAL METHODS
A multidomain spectral code [36] is used for our 3Dsimulation. The domain decomposition is based on thedefinition of a Schur complement and implemented withMPI library. Incompressibility of the flow is enforced bythe projection-correction method. The spatial discretizationis performed with the spectral collocation method, while thetemporal discretization is carried out through a second-orderimplicit-explicit mixed scheme. The overall procedure consistsof solving several general Helmholtz equations.
1. Discretization
The spectral collocation method is used for spatial dis-cretization. In the horizontal direction x, Chebyshev modesare used. In the transverse direction y, Fourier modes areused, as the periodic boundary condition is imposed. Thedomain is divided into four subdomains along the verticaldirection z. Chebyshev modes are used for each subdomainin this direction. A periodic communicator is defined throughMPI to enforce the periodic vertical boundary condition overthe full domain. The solution of Navier-Stokes equations inthe interior of each subdomain is executed independentlyon a single processor. The continuity of variables and theirfirst derivatives across the interface between the subdo-mains is ensured by defining a Schur complement method,where the Schur matrix is built using an influence matrixtechnique [36].
A second-order mixed explicit-implicit scheme is adoptedfor the temporal integration. The diffusive term is treatedimplicitly, while the convective term is calculated explicitly.The discretized Navier-Stokes equations can be recast into four
general Helmholtz equations for u, v, w, θ , respectively,Pr√Ra
∇2un+1 − 3
2�tun+1
= ∂pn+1
∂x− 4un − un−1
2�t+ 2(V · �∇u)n − (V · �∇u)n−1,
(A1)Pr√Ra
∇2vn+1 − 3
2�tvn+1
= ∂pn+1
∂y− 4vn − vn−1
2�t+ 2(V · �∇v)n − (V · �∇v)n−1,
(A2)
Pr√Ra
∇2wn+1 − 3
2�twn+1
= ∂pn+1
∂z− 4wn − wn−1
2�t+ 2(V · �∇w)n − (V · �∇w)n−1,
(A3)
1√Ra
∇2θn+1 − 3
2�tθn+1
= −4θn − θn−1
2�t+ 2(V · �∇θ )n − (V · �∇θ )n−1. (A4)
A fifth general Helmholtz equation for a potential needsto be solved in the procedure of the projection-correctionmethod to ensure the incompressibility of the flow. Allfive general Helmholtz problems are solved by use ofthe matrix-diagonalization method, which is presented asfollows.
2. Solution of the general Helmholtz problem
The general 3D Helmholtz problem reads as
(∇2 − λ)f = S, (A5)
where ∇2 = ∂2
∂x2 + ∂2
∂y2 + ∂2
∂z2 ; λ = 3√
Ra2Pr�t
for u,v,w; λ = 3√
Ra2�t
for θ ; and S represents the source term. The idea solvethis equation in the discrete form is to invert the operator
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TRANSITION TO CHAOS OF NATURAL CONVECTION . . . PHYSICAL REVIEW E 88, 023010 (2013)
(∇2 − λ). In our code, the matrix-diagonalization methodis employed [36,47,48]. In fact, the second derivative, forexample ∂2
∂x2 , in the discrete form constitutes a regular matrix,which is diagonalizable and invertible, so we have D2
x =P�xP
−1, D2y = Q�yQ
−1, D2z = R�zR
−1, where �x , �y ,�z are diagonal matrix containing the eigenvalues, and thematrices P , Q, R are formed by the eigenvectors of D2
x ,D2
y , D2z , respectively. In the discrete forms, the 3D Helmholtz
equation takes the following form:(Iz ⊗ Iy ⊗ D2
x + Iz ⊗ D2y ⊗ Ix + D2
z ⊗ Iy ⊗ Ix
− Iz ⊗ Iy ⊗ Ixλ)F = S, (A6)
where ⊗ is the Kronecker product operator. Multi-plying Eq. (A6) by (P −1 ⊗ Q−1 ⊗ R−1), we can pass
this equation into the eigenspace in the following way.For the first term on the left side of Eq. (A6),we have (R−1 ⊗ Q−1 ⊗ P −1)(Iz ⊗ Iy ⊗ D2
x) = (Iz ⊗ Iy ⊗�x)(R−1 ⊗ Q−1 ⊗ P −1) by using twice the property (A ⊗B)(C ⊗ D) = AC ⊗ BD. With similar treatment for the otherterms, we can obtain
(Iz ⊗ Iy ⊗ �x + Iz ⊗ �y ⊗ Ix + �z ⊗ Iy ⊗ Ix
− Iz ⊗ Iy ⊗ Ixλ)F = S, (A7)
where F = (R−1 ⊗ Q−1 ⊗ P −1)F and S = (R−1 ⊗ Q−1 ⊗P −1)S. Therefore, to solve the 3D Helmholtz problem, we firstmultiply the source term S by (R−1 ⊗ Q−1 ⊗ P −1) to obtainS. Then F can be easily obtained by inverting the operator infront of F in Eq. (A7). Finally, multiplying F by P ⊗ Q ⊗ R,we get the solution F .
APPENDIX B: OPERATORS IN MULTISCALE ANALYSIS SECTION
The operators defined in Eqs. (15) to (17) can be expressed as
M0 =[∇2
0 0
0 1
], (B1)
M1 =[
2 ∂∂z0
∂∂z1
0
0 0
], (B2)
M2 =[
∂2
∂z21
0
0 0
], (B3)
L0 =⎡⎣ Pr√
Rac∇4
0 − W ∂∂z0
∇20 + ∂2W
∂x2∂
∂z0Pr ∂
∂x
∂�∂x
∂∂z0
1√Rac
∇20 − W ∂
∂z0
⎤⎦ , (B4)
L1 =[
4 Pr√Rac
∂∂z0
∂∂z1
∇20 − W ∂
∂z1∇2
0 − 2W ∂∂z1
∂2
∂z20+ ∂2W
∂x2∂
∂z10
∂�∂x
∂∂z1
2√Rac
∂∂z0
∂∂z1
− W ∂∂z1
], (B5)
L2 =⎡⎣ 2 Pr√
Rac
∂2
∂z21
(∇20 + 2 ∂2
∂z20
) − 3W ∂∂z0
∂2
∂z21
0
0 1√Rac
∂2
∂z21
⎤⎦
+⎡⎣− 1
2Pr√Ra3
c
∇40 − W
2Rac
∂∂z0
∇20 + 1
2Rac
∂2W∂x2
∂∂z0
0
0 − 12
1√Ra3
c
∇20 − W
2Rac
∂∂z0
⎤⎦ , (B6)
b0,ψ (φα,φβ) =(
∂ψα
∂z0
∂
∂x− ∂ψα
∂x
∂
∂z0
)∇2
0ψβ, (B7)
b0,θ (φα,φβ) =(
∂ψα
∂z0
∂
∂x− ∂ψα
∂x
∂
∂z0
)θβ, (B8)
b1,ψ (φα,φβ) =(
∂ψα
∂z1
∂
∂x− ∂ψα
∂x
∂
∂z1
)∇2
0ψβ + 2
(∂ψα
∂z0
∂
∂x− ∂ψα
∂x
∂
∂z0
)∂
∂z0
∂
∂z1ψβ, (B9)
b1,θ (φα,φβ) =(
∂ψα
∂z1
∂
∂x− ∂ψα
∂x
∂
∂z1
)θβ, (B10)
where ∇0 = ∂2
∂x2 + ∂2
∂z20, Rac is the critical Rayleigh number found in the linear stability analysis, W is the vertical velocity of the
base flow, and � is the temperature of the base flow.
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[1] P. Manneville, Dyn. Spatio-Temp. Cellular Struct. 207, 41(2006).
[2] J. Elder, J. Fluid Mech. 23, 77 (1965).[3] C. Vest and V. Arpaci, J. Fluid Mech. 36, 1 (1969).[4] Y. Oshima, J. Phys. Soc. Jpn. 30, 872 (1971).[5] G. K. Batchelor, Quart. Appl. Math. 12, 209 (1954).[6] A. Gill and A. Davey, J. Fluid Mech. 35, 775 (1969).[7] R. Bergholz, J. Fluid Mech. 84, 743 (1978).[8] P. Daniels and M. Weinstein, J. Eng. Math. 24, 179 (1989).[9] J. Cornet and C. Lamarque, Mech. Res. Comm. 24, 179 (1997).
[10] J. Mizushima and K. Gotoh, J. Phys. Soc. Jpn. 52, 1206 (1983).[11] J. Mizushima and Y. Saito, Fluid Dyn. Res. 2, 183 (1987).[12] J. Mizushima and H. Tanaka, Phys. Fluids 14, L21 (2002).[13] J. Mizushima and H. Tanaka, J. Phys. Soc. Jpn. 71, 2898 (2002).[14] S. Xin, Ph.D. thesis, Universite Paris VI, 1993.[15] D. R. Chenoweth and S. Paolucci, Phys. Fluids 28, 2375 (1985).[16] D. R. Chenoweth and S. Paolucci, J. Fluid Mech. 169, 173
(1986).[17] P. L. Quere, ASME J. Heat Transf. 112, 965 (1990).[18] S. Wakitani, Phys. Fluids. 10, 1924 (1998).[19] H. Gunes, Fluid Dyn. Res. 30, 1 (2002).[20] B. Lartigue, S. Lorente, and B. Bourret, Int. J. Heat Mass Transf.
43, 3157 (2000).[21] J. L. Wright, H. Jin, K. G. T. Hollands, and D. Naylor, Int. J.
Heat Mass Transf. 49, 889 (2006).[22] A. Chait and S. Korpela, J. Fluid Mech. 200, 189 (1989).[23] M. Nagata and F. Busse, J. Fluid Mech. 135, 1 (1983).[24] R. M. Clever and F. H. Busse, Chaos Soliton. Fract. 5, 1795
(1995).[25] A. Suslov and S. Paolucci, Fluid Dyn. Res. 35, 159 (2004).[26] A. Suslov and S. Paolucci, J. Fluid Mech. 344, 1 (1997).[27] D. A. Bratsun, A. V. Zyuzgin, and G. F. Putin, Int. J. Heat Fluid
Flow 24, 835 (2003).[28] J. R. Phillips, Int. J. Heat Mass Transf. 39, 2485 (1996).[29] J. Elder, J. Fluid Mech. 23, 99 (1965).[30] T. Versteegh and F. Nieuwstadt, Int. J. Heat Fluid Flow 19, 135
(1998).
[31] T. Versteegh and F. Nieuwstadt, Int. J. Heat Mass Transf. 42,3673 (1999).
[32] M. Wang, S. Fu, and G. Zhang, Phys. Rev. E 66, 066306(2002).
[33] P. Hall, J. Fluid Mech. 703, 99 (2012).[34] A. Sergent and P. L. Quere, J. Phys. Conf. Ser. 318, 082010
(2011).[35] B. Podvin and A. Sergent, Phys. Fluids 23, 105106 (2012).[36] S. Xin, J. Chergui, and P. L. Quere, in Parallel Computa-
tional Fluid Dynamics 2008, edited by D. Tromeur-Dervout,G. Brenner, D. Emerson, and J. Erhel, Lecture Notes inComputational Science and Engineering Vol. 74 (Springer,Berlin, 2008), pp. 163–172.
[37] D. W. Ruth, Int. J. Heat Mass Transf. 22, 1199 (1979).[38] C. Cossu, Lecture Notes, Ecole Polytechnique, Palaiseau, France
(2009).[39] J. Tao and F. Zhuang, Phys. Rev. E 62, 7957 (2000).[40] R. Henderson and D. Barkley, Phys. Fluids 8, 1683
(1996).[41] J. C. R. Hunt, A. A. Wray, and P. Moin, in Studying Tur-
bulence Using Numerical Simulation Databases, 2, Vol. 1(Center for Turbulence Research, Stanford University, 1988),pp. 193–208.
[42] C. Ho and P. Huerre, Ann. Rev. Fluid Mech. 16, 365 (1984).[43] J. Maurer and A. Libchaber, Le J. Phys. Lett. 40, 419
(1979).[44] M. Giglio, S. Musazzi, and U. Perini, Phys. Rev. Lett. 47, 243
(1981).[45] D. Armbruster, J. Guckenheimer, and P. Holmes, Physica D 29,
257 (1987).[46] I. Melbourne, P. Chossat, and M. Golubitsky, Proc. R. Soc. Edin.
A Math. 113, 315 (1989).[47] G. Labrosse, Methodes spectrales: methodes locales, methodes
globales, problemes d’Helmoltz et de Stokes, equations deNavier-Stokes (Ellipses, 2011), Chap. 12, pp. 168–170.
[48] P. Haldenwang, G. Labrosse, S. Abboudi, and M. Deville,J. Comput. Phys. 55, 115 (1984).
023010-18